From 097f244236b682cc77c6ecf25be4150091d8daf9 Mon Sep 17 00:00:00 2001
From: Norbert Preining <norbert@preining.info>
Date: Sun, 2 Oct 2022 03:04:14 +0000
Subject: CTAN sync 202210020304

---
 info/mathtrip/src/calc3.tex | 58 +++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 58 insertions(+)
 create mode 100644 info/mathtrip/src/calc3.tex

(limited to 'info/mathtrip/src/calc3.tex')

diff --git a/info/mathtrip/src/calc3.tex b/info/mathtrip/src/calc3.tex
new file mode 100644
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+++ b/info/mathtrip/src/calc3.tex
@@ -0,0 +1,58 @@
+%This command produces the text of the calculus in the first column
+%of the first horizontal area
+%
+%The command has one parameter:
+%        1) The width of the mathematical text
+\newcommand\TEightCalculusThree[1]{%
+   %Command to put a strut in these formulae
+   \def\TEightCalcC{\rule[-5ex plus .5ex minus 1ex]{0pt}{6ex plus 2ex minus 1ex}}
+   %A much bigger strut
+   \def\TEightCalcBig{\rule[-5ex plus .5ex minus 1ex]{0pt}{12ex plus 2ex minus 1ex}}
+   \parbox[t]{#1}{%
+      \TEightSeriesCalculusFontSize
+      \begin{DisplayFormulae}{62}{\SpaceBeforeFormula}{\TEightBaselineSkipFormulae}{\BigChar}{\StyleBold}
+         %Formula 62
+         \Fm{\int \frac{dx}{x \sqrt{x^2 - a^2}} = \frac{1}{a} \arccos \frac{a}{\vert x\vert}\quad a > 0}
+         \Fm{\int \frac{dx}{x^2 \sqrt{x^2 \pm a^2}} = \mp \frac{\sqrt{x^2 \pm a^2}}{a^2 x}}
+         \Fm{\int \frac{\xdx}{\sqrt{x^2 \pm a^2}} = \sqrt{x^2 \pm a^2}}
+         \Fm{\int \frac{\sqrt{x^2 \pm a^2}}{x^4} \, dx = \mp \frac{(x^2 + a^2)^{3/2}}{3a^2 x^3}}
+         \Fm{\int \frac{dx}{a x^2  + bx + c} = \left\{
+             \begin{array}{lr}
+                 \displaystyle \frac{1}{\sqrt{b^2 -4ac}} \ln \left\vert\frac{2ax + b - \sqrt{b^2 -4ac}}{2ax + b + \sqrt{b^2 -4ac}}\right\vert 
+                     &\text{if $b^2 > 4ac$\TEightCalcC} \\
+                 \displaystyle \frac{2}{\sqrt{4ac - b^2}} \arctan \frac{2ax + b}{\sqrt{4ac - b^2}} 
+                 &\text{if $b^2 < 4ac$\TEightCalcC} \\
+             \end{array}\right.
+         }
+         \Fm{\int \frac{dx}{\sqrt{a x^2  + bx + c}}  = \left\{
+             \begin{array}{lr}
+                  \displaystyle \frac{1}{\sqrt{a}} \ln \left\vert 2ax + b + 2\sqrt{a} \sqrt{ax^2 + bx + c}\right\vert 
+                   &\text{if $a > 0$\TEightCalcC}\\
+                   \displaystyle \frac{1}{\sqrt{- a}} \arcsin \frac{-2ax - b}{\sqrt{b^2 - 4ac}} 
+                   &\text{if $a < 0$\TEightCalcC}\\
+              \end{array}\right.
+         }
+         \def\FirstPart{\int \sqrt{a x^2  + bx + c} \, dx =}
+         \FmPartA{\FirstPart}
+         \FmPartB{\FirstPart}{\frac{2ax + b}{4a} \sqrt{a x^2  + bx + c} + 
+             \frac{4ax - b^2}{8a} \int \frac{dx}{\sqrt{a x^2  + bx + c}}}
+         \Fm{\int \frac{\xdx}{\sqrt{a x^2  + bx + c}} = \frac{\sqrt{a x^2  + bx + c}}{a} - \frac{b}{2a} 
+              \int \frac{dx}{\sqrt{a x^2  + bx + c}}}
+        \Fm{\TEightCalcBig\int \frac{dx}{x \sqrt{a x^2  + bx + c}} =  \left\{
+            \begin{array}{l@{\hspace{.3ex plus .3ex minus .1ex}}r}
+                 \displaystyle \frac{-1}{\sqrt{c}} \ln \left\vert\frac{2\sqrt{c} \sqrt{ax^2 + bx + c} + bx + 2c}{x}\right\vert 
+                  &\text{\small if $c > 0$}\\
+                  \displaystyle \frac{1}{\sqrt{- c}} \arcsin \frac{bx + 2c}{\vert x \vert\sqrt{b^2 - 4ac}} 
+                   &\text{\small if $c < 0$} \\
+            \end{array}\right.
+        }
+        \Fm{\int x^3  \sqrt{x^2 + a^2} \, dx = (\frac{1}{3} x^2 - \frac{2}{15} a^2)(x^2 + a^2)^{3/2}}
+        %Formula 72
+        \Fm{\int x^n \sin (ax) \, dx = - \frac{1}{a} x^n \cos (ax) + \frac{n}{a} \int x^{n-1} \cos (ax) \, dx} 
+        \Fm{\int x^n \cos (ax) \, dx = \frac{1}{a} x^n \sin (ax) - \frac{n}{a} \int x^{n-1} \sin (ax) \, dx} 
+          \Fm{\int x^n e^{ax} \, dx = \frac{x^n e^{ax}}{a} - \frac{n}{a} \int x^{n-1} e^{ax} \, dx}
+        \Fm{\int x^n \ln (ax) \, dx = x^{n+1}\left(\frac{\ln (ax)}{n+1} - \frac{1}{(n+1)^2}\right)}
+        \Fm{\int x^n (\ln ax)^m \, dx = \frac{x^{n+1}}{n+1}(\ln ax)^m - \frac{m}{n+1}\int x^n (\ln ax)^{m-1} \, dx}
+      \end{DisplayFormulae}
+   }
+}
-- 
cgit v1.2.3

